Aircraft: B757
Phase of flight: Initial Climb
Anomaly: Aircraft Equipment Problem Critical; Deviation - Speed All Types; Deviation / Discrepancy - Procedural Published Material / Policy; Flight Deck / Cabin / Aircraft Event Smoke / Fire / Fumes / Odor

APU fire return to field. Boarding and pre-flight normal. Once the packs were turned on the cabin cooled down easily; but we were not able to get the flight deck under 80 degrees until after engine start. Taxi was also normal; from [the gate] onto Taxiway 1; Taxiway 2; and to RWY XXL. To fly the departure up to and maintain 3;000 ft. Takeoff normal (full thrust takeoff per Maintenance request on maintenance release). On the climb out approaching 3;000 ft; in LNAV; we hit ALT CAP (Altitude Capture); and PNF (myself) rolled the speed to 230 knots. Right at this moment the APU FIRE EICAS message displayed; and the Fire bell alert sounded. Captain took both flight controls and radios; put the auto pilot on and addressed ATC. [Advised ATC] and requested an immediate return to the field for landing. I ran the APU QRC paper checklist. Which reads; Pull the APU Fire handle and rotate to the stop for 1 second. I completed the QRC items; and then turned to the electronic checklist to complete the full APU FIRE checklist. Which stated 'Choose one: APU Fire warning light stays illuminated (which it did); land at nearest suitable airport'. Captain then handed flight controls and radios to me; so that he could consult with the Flight Attendants and the Company. After doing so he took back over the role of PF (Pilot Flying). Captain then made a turn with ATC instruction back toward field for a downwind to RWY XXL. Still level at 3;000 ft. ATC requested where the fire indication was. We set up for an ILS approach; with appropriate frequency and FMS arrival selection for XXL. We were then given a descent to 2;000 ft. by ATC as well as a turn to base leg. Captain began to slow to configure for landing. Using speedbrakes to slow the aircraft. On this turn in the descent with Flaps 1; while retracting speedbrakes; and the MCP (Mode Control Panel) speed in the window set to minimum flaps 1 speed; we had a momentary shaker activation; less than 1 second; and I immediately put out Flaps 5 and we continued to slow and configure from there. We were cleared to maintain 2;000 ft. until established; and cleared for the ILS approach. We continued to add flaps to 15; then 20. Gear down; flaps 25 and 30 without event. Landing data was requested for landing on XXL; overweight 235k lbs; Auto brakes 2 was set for sufficient stopping distance. Also performed a small dip low on GS for touchdown to use full runway length. We touched down with 200 fpm descent. Rolled to Taxiway 3. Exited runway at 60 kts. Continued onto Taxiway 2 to hold short of Taxiway 4. At this time the [it was] assessed that no heat was found in the tail or any other part of the aircraft. The flight attendants also assessed no smoke or heat in the cabin. The decision was then made to taxi to a gate expeditiously to deplane passengers and get maintenance on board. Taxied to the gate; shut down the left engine to allow the jet bridge to be attached to the airplane; then shut down the right engine once ground power was established. APU Fire Indication after takeoff. Flight was the first segment on day one from ZZZ to ZZZ1. I was the flying pilot for this leg. Loading and boarding were normal with a mostly full flight. The cabin of the aircraft had been cooled nicely; but we were having a bit of a challenge cooling down the flight deck; with temperatures hovering in the high 80's. The ZZZ weather was 340/10; 10SM FEW 040 SCT080 BKN250 28/18 29.86. We pushed off the gate on time. [We taxied] to Runway XXL and contacted Tower. We accomplished the before takeoff checklist items and were cleared to takeoff Runway XXL on the ZZZ Departure. The maintenance release noted that a Max power takeoff was due; so we did so; at a TOG weight of 238;192 lbs and the flaps 5. During takeoff roll as we accelerated; water dripped out of the overhead panels which was interesting; we took off; cleaned up the gear and flaps as scheduled. Somewhere between 2;000 ft and 3;000 ft. in the climb in a left hand turn to heading 290; we had a Master Warning and an EICAS annunciation APU Fire. I continued to fly the aircraft; took the radios; and while the First Officer accomplished the APU Fire QRC. It is important to note that the fire light remained on and did not go out after running the QRC with no other indications of a fire. Also; it is important to note that the APU was shut down on the ground after engine start during the after-start checklist; and had been off for approximately 20 minutes. We ran the appropriate checklists; notified Dispatch and spoke with the flight attendants. The Purser was informed that we had an APU Fire indication and was asked if there were any fire indications in the aft cabin? They said they did not have smoke but noted a slight odor (possibly the fire extinguishing agent that was fired into the APU during the QRC). I told them to prepare the cabin quickly as we would be on the ground in 8 minutes. My FO (First Officer) and I spoke about the overweight landing; briefed the approach; and identified the ILS. We planned a flaps 30 ILS to Runway XXL at a weight of 235;000 lbs. as 235;000 lbs. was below our performance limit weight of 259.9; so a 30-flap landing was appropriate. Fortunately; we discussed the possibility of an overweight landing prior to takeoff; so we were already familiar with the requirements. Although we suspected an erroneous APU Fire warning; time was still a driving factor as the fire indication never went away. We maintained 250 kts for as long as possible then configured using flaps and speed brakes to slow the aircraft. We were configured with flaps 1 and were decelerating with the flaps tracking to 5 and the speed brakes out. We decelerated to flaps 5 maneuvering speed; and just as I started to retract the speed brakes for literally a split-second I heard a sound that was so brief I could not tell what it was. The FO told me that it was a split second of the shaker due to the boards being out at the flaps 5 maneuvering speed. We continued configuring with gear and flaps 30; accomplished the landing checklist. We flew the ILS to Runway XXL with the autopilot and auto throttles engaged until 500 ft when I disconnected the autopilot. I disengaged the auto throttles around 100 ft. I briefed that I would slightly duck under the glideslope with a planned touched down early in the landing zone.We touched down approximately 1;200 ft. down the runway with auto brakes 2 selected. The touchdown was extremely smooth with a VVI (Vertical Velocity Indicator) of less than 200 fpm. During the landing roll; we queried the Tower to see if they could see any smoke trailing the aircraft; and they said all looked normal. We rolled clear of the runway on Taxiway 3 and stopped on Taxiway 4 where the ground personnel looked us over. The ground personnel informed us that they could see no indications of a fire and that their thermal imaging scan showed no heat in the APU area. We ran through our after-landing flows and taxied to the gate. We shut downthe left engine so they could start deplaning the passengers. When ground power became available we secured the right engine and secured the aircraft. I proceeded into the cabin to chat with maintenance; passengers etc. Maintenance opened the APU area and said that they suspected a faulty fire loop; as there was no evidence of a fire. Maintenance later found that a faulty APU fire control panel was the cause of the warning. Also; the total flight time from wheels up to wheels down was 11 minutes. That is extremely fast when considering all the factors to evaluate. All in all; I feel that this was a well-executed event; and resulted in the safe return for passengers and crew. B757 flight crew reported an APU fire warning after takeoff and returned to the departure airport.